Compactness theorems of gradient Ricci solitons

Abstract

In this paper, we prove the compactness theorem for gradient Ricci solitons. Let $(M_{\alpha },g_{\alpha })$ be a sequence of compact gradient Ricci solitons of dimension $n\ge 4$, whose curvatures have uniformly bounded $L^{\frac{n}{2}}$ norms, whose Ricci curvatures are uniformly bounded from below with uniformly lower bounded volume and with uniformly upper bounded diameter, then there must exists a subsequence $(M_{\alpha },g_{\alpha })$ converging to a compact orbifold $(M_{\infty },g_{\infty })$ with finitly many isolated singularities, where $g_{\infty }$ is a gradient Ricci soliton metric in an orbifold sense.

21pages

References in corpus (1)

• The compactness result for Kähler Ricci solitons

Cited by in corpus (14)

• Geometry of shrinking Ricci solitons
• A compactness theorem for complete Ricci shrinkers
• Space of Ricci flows (II)
• Compactness theory of the space of Super Ricci flows
• Structure theory of non-collapsed limits of Ricci flows
• $\epsilon$-Regularity and Structure of 4-dimensional Shrinking Ricci Solitons
• Degeneration of Kähler-Ricci solitons
• The Curvature of Gradient Ricci Solitons
• Bubble-Tree Convergence and Local Diffeomorphism Finiteness for Gradient Ricci Shrinkers
• $\epsilon$-regularity for shrinking Ricci solitons and Ricci flows
• Degeneration of shrinking Ricci solitons
• Convergence of compact Ricci solitons
• The Rigidity of Ricci shrinkers of dimension four
• On the structure of Ricci shrinkers